Pose from Fundamental matrix and vice versa

Question

I have computed the Fundamental Matrix between two cameras using opencv's findFundamentalMat. Then I plot the epipolar lines in the image. And I get something like:

Now, I tried to get the pose from that fundamental matrix, computing first the essential matrix and then using Hartley & Zissserman approach.

K2=np.mat(self.calibration.getCameraMatrix(1))
K1=np.mat(self.calibration.getCameraMatrix(0))
E=K2.T*np.mat(F)*K1

w,u,vt = cv2.SVDecomp(np.mat(E))   
if np.linalg.det(u) < 0:
    u *= -1.0
if np.linalg.det(vt) < 0:
    vt *= -1.0 
#Find R and T from Hartley & Zisserman
W=np.mat([[0,-1,0],[1,0,0],[0,0,1]],dtype=float)
R = np.mat(u) * W * np.mat(vt)
t = u[:,2] #u3 normalized.

In order to check everything until here was correct, I recompute E and F and plot the epipolar lines again.

S=np.mat([[0,-T[2],T[1]],[T[2],0,-T[0]],[-T[1],T[0],0]])
E=S*np.mat(R)
F=np.linalg.inv(K2).T*np.mat(E)*np.linalg.inv(K1)

But surprise, the lines have moved and they don't go through the points anymore. Have I done something wrong?

It might be related with this question http://answers.opencv.org/question/18565/pose-estimation-produces-wrong-translation-vector/, but they didn't provide a solution

The matrices I get are:

Original F=[[ -1.62627683e-07  -1.38840952e-05   8.03246936e-03]
 [  5.83844799e-06  -1.37528349e-06  -3.26617731e-03]
 [ -1.15902181e-02   1.23440336e-02   1.00000000e+00]]

E=[[-0.09648757 -8.23748182 -0.6192747 ]
 [ 3.46397143 -0.81596046  0.29628779]
 [-6.32856235 -0.03006961 -0.65380443]]

R=[[  9.99558381e-01  -2.72074658e-02   1.19497464e-02]
  [  3.50795548e-04   4.12906861e-01   9.10773189e-01]
  [ -2.97139627e-02  -9.10366782e-01   4.12734058e-01]]

T=[[-8.82445166e-02]
 [8.73204425e-01]
 [4.79298380e-01]]

Recomputed E=
[[-0.0261145  -0.99284189 -0.07613091]
 [ 0.47646462 -0.09337537  0.04214901]
 [-0.87284976 -0.01267909 -0.09080531]]

Recomputed F=
[[ -4.40154169e-08  -1.67341327e-06   9.85070691e-04]
 [  8.03070680e-07  -1.57382143e-07  -4.67389530e-04]
 [ -1.57927152e-03   1.47100268e-03   2.56606003e-01]]

Schoolroom

Answer 1

The first F is defined up to scale, hence if you're going to compare the returned F and with the F matrix computed from E you need to normalize them to make sure both are at the same scale. Hence you need to normalize the second computed F.

Situla

I don't aim to compare numerically both but just the epipolar lines they plot... Is it important then? When normalizing do you mean column normalization? – Schoolroom

Yes in plotting epipolar lines you must be using OpenCV's function and it also requires normalized matrix according to my understanding. You have to read a more about normalization in OpenCV. (I am not very sure about the type of normalization as I used it long back.) – Situla